To explore the doping dependence of the recently discovered charge-density-wave (CDW) order in YBa2Cu3Oy, we present a bulk-sensitive high-energy x-ray study for several oxygen concentrations, including strongly underdoped YBa2Cu3O6.44. Combined with previous data around the so-called 1/8 doping, we show that bulk CDW order exists at least for hole concentrations (p) in the CuO2 planes of 0.078 less than or similar to p less than or similar to 0.132. This implies that CDW order exists in close vicinity to the quantum critical point for spin-density-wave (SDW) order. In contrast to the pseudogap temperature T*, the onset temperature of CDW order decreases with underdoping to T-CDW similar to 90 K in YBa2Cu3O6.44. Together with a weakened order parameter this suggests a competition between CDW and SDW orders. In addition, the CDW order in YBa2Cu3O6.44 shows the same type of competition with superconductivity as a function of temperature and magnetic field as samples closer to p = 1/8. At low p the CDW incommensurability continues the previously reported linear increasing trend with underdoping. In the entire doping range the in-plane correlation length of the CDW order in b axis direction depends only very weakly on the hole concentration, and appears independent of the type and correlation length of the oxygen-chain order. The onset temperature of the CDW order is remarkably close to a temperature T-dagger that marks the maximum of 1/(T1T) in planar Cu-63 NQR/NMR experiments, potentially indicating a response of the spin dynamics to the formation of the CDW. Our discussion of these findings includes a detailed comparison to the charge stripe order in La2-xBaxCuO4.